Question

In Problems 1–10 write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L.

$\mathbf{2}\mathit{y}\mathbf{"}\mathbf{-}\mathbf{3}\mathit{y}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{=}\mathbf{1}$

Short answer

$\left(2D+1\right)\left(D-2\right)y=1$

Step-by-step solution

Use the Differential linear operators to find the solution

We have the non-homogeneous second order differential equation,

$2y"-3y\text{'}-2y=1$

And we have to write it in the form using the differential linear operators

as

$$\begin{gathered} 2D^{2}y-3Dy-2y=1 \\ \left(2D^2-3D-2\right)y=1 \end{gathered}$$

After that we have to factorize it as,

$\left(2D+1\right)\left(D-2\right)y=1$

Where g (x) = 1.

Hence the solution isrole="math" localid="1667821820021" $\left(2D+1\right)\left(D-2\right)\mathit{y}\mathbf{=}\mathbf{1}$